Question

The waiting time for a bus at a certain bus stop has a uniform distribution over the interval from 0 to 25 minutes. (a) What is the probability that a person has to wait less than 6 minutes for the bus? (b) What is the probability that a person has to wait between 10 and 20 minutes for the bus?

Answer 1

(a) The probability that a person has to wait less than 6 minutes for the bus is 0.24.

(b) The probability that a person has to wait between 10 and 20 minutes for the bus is 0.40.

Explanation:

Let The random variable X be defined as the waiting time for a bus at a certain bus stop.

The random variable X follows a continuous Uniform distribution with parameters a = 0 and b = 25.

The probability density function of X is:

`f_(X)(x)=\left \{ {{(1)/(b-a);\ a<X<b;\ a<b} \atop {0;\ otherwise}} \right.`

(a)

Compute the probability that a person has to wait less than 6 minutes for the bus as follows:

`P(X<6)=\int\limits^(6)_(0){(1)/(25-0)}\, dx`

`=(1)/(25)* \int\limits^(6)_(0){1}\, dx`

`=(1)/(25)* [x]^(6)_(0)`

`=(1)/(25)* [6-0]`

`=0.24`

Thus, the probability that a person has to wait less than 6 minutes for the bus is 0.24.

(b)

Compute the probability that a person has to wait between 10 and 20 minutes for the bus as follows:

`P10<(X<20)=\int\limits^(20)_(10){(1)/(25-0)}\, dx`

`=(1)/(25)* \int\limits^(20)_(10){1}\, dx`

`=(1)/(25)* [x]^(20)_(10)`

`=(1)/(25)* [20-10]`

`=0.40`

Thus, the probability that a person has to wait between 10 and 20 minutes for the bus is 0.40.